Year 11 Extended · Unit 4 Rational expressions and functions · Lesson 2 of 7

How do you multiply or divide fractions that have letters in them?

Algebraic fractions multiply and divide exactly as numeric fractions do. This lesson factorizes every part first, cancels across the multiplication, and asks what the shorter answer might not show.

FunctionsNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7: work out 2⁄3 ÷ 4⁄9.
    Answer3⁄2. Multiply by the reciprocal: 2/3 × 9/4 = 18/12.
  2. From lesson 1: simplify (3x + 9) ÷ (x2 − 9).
    Answer3 ÷ (x − 3).
  3. From Year 9: factorize x2 − 6x + 8.
    Answer(x − 2)(x − 4).
Words used in this lesson
reciprocal
the fraction turned upside down. For example: the reciprocal of 3/7 is 7/3.
divisor
the fraction you divide by. For example: the second fraction in a ÷ b.
quotient
the result of a division. For example: 12 ÷ 4.
cancel across
remove a factor from one top and another bottom. For example: the 5 in 5/9 × 4/5.
excluded value
a value of x that has to be left out. For example: a zero of a bottom.

What goes where when letters get involved?

Tomás multiplies (x2 − 4x) ÷ (x + 2) by (x + 2) ÷ (x2 − 16). He factorizes every part, cancels, and gets a much shorter fraction.

Predict first: which values of x are ruled out by the original product, and does the shorter fraction show all of them? Use the figure to try several values of x, then press the divide button and look again.

1
0.2the result
0.2the cancelled form

Multiply, x = 1: left −1, right −0.2, so the result is 0.2 and the cancelled form gives 0.2.

Show me the steps
  1. Multiply, x = 1: left = −3 ÷ 3 = −1, right = 3 ÷ (−15) = −0.2, product 0.2. The cancelled form x ÷ (x + 4) gives 1 ÷ 5 = 0.2.
  2. Multiply, x = 4: the right fraction has denominator 16 − 16 = 0, so the product has no value. The cancelled form still gives 4 ÷ 8 = 0.5, so x = 4 must be stated as excluded.
  3. Divide, x = 1: −1 ÷ −0.2 = 5. The cancelled form x(x − 4)2(x + 4) ÷ (x + 2)2 also gives 5.
  4. Divide, x = −2: the left denominator is 0, and the right fraction (the divisor) is 0 on top. Dividing by a zero is not allowed, so there is no value.

Factorize, flip, cancel across. To multiply, factorize every top and bottom, cancel any factor that is on the top of one fraction and the bottom of another, then multiply what is left. To divide, turn the second fraction upside down and multiply.

Excluded values come from every denominator in the question, and also from the divisor's top, because the divisor must not be zero. They are read before cancelling.

Flip only the second fraction. In a ÷ b, turn b upside down and leave a alone. Check with numbers: 1⁄2 ÷ 1⁄4 = 2, but flipping the first fraction instead gives 2⁄1 × 1⁄4 = 1⁄2.

Worked examples

1. Multiply (x2 − 9) ÷ (2x) by (4x) ÷ (x + 3).

  1. Factorize: (x − 3)(x + 3) ÷ (2x) × 4x ÷ (x + 3).
  2. Cancel x + 3 and x, and divide 4 by 2: 2(x − 3), with x ≠ 0 and x ≠ −3.

2. Divide (x2 − 1) ÷ (x + 3) by (x + 1) ÷ (x2 + 3x).

  1. Flip the second fraction and factorize: (x − 1)(x + 1) ÷ (x + 3) × x(x + 3) ÷ (x + 1).
  2. Cancel x + 1 and x + 3: x(x − 1). Excluded: x = −3 and x = 0 (bottoms) and x = −1 (the divisor's top).

3. Simplify ((x2 + 5x + 6) ÷ (x2 − 4)) ÷ ((x + 3) ÷ (x − 2)).

  1. Factorize and flip: (x + 2)(x + 3) ÷ ((x − 2)(x + 2)) × (x − 2) ÷ (x + 3).
  2. Everything cancels: 1, for every x except −3, −2 and 2.

Your go. The first step is given. Simplify (x2 − 16) ÷ (x + 2) × (3x + 6) ÷ (x + 4). The top is (x − 4)(x + 4).

  1. Factorize 3x + 6.
  2. Cancel across the multiplication and state the result.
Check the answer

3x + 6 = 3(x + 2), so x + 2 and x + 4 cancel and the result is 3(x − 4), with x ≠ −2 and x ≠ −4.

Check before you go on

1. Which is the simplest form of 6x ÷ (x2 − 25) ÷ (3x ÷ (x − 5))?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. (6x) ÷ (x + 4) × (x2 − 16) ÷ (3x2) simplifies to a(x − b) ÷ x. Find a + b.

3. Simplify (x2 − 4) ÷ (x + 5) ÷ ((x − 2) ÷ (x2 + 5x)), then evaluate at x = 3.

Level 2 · Building

4. (x2 + 3x) ÷ (x2 − 25) ÷ ((x + 3) ÷ (x + 5)) equals 2. Find x.

5. How many values of x cannot be used in (x + 1) ÷ (x − 2) ÷ ((x − 4) ÷ (x + 3))?

Level 3 · Stretch

6. (x2 + kx − 15) ÷ (x − 3) × (x + 2) ÷ (x + 5) simplifies to x + 2. Find k.

7. A cooperative in Kerala has (x + 3) ÷ (x − 1) tonnes of rice and packs it into sacks that each hold (x + 3) ÷ (x2 − 1) tonnes. How many sacks does it fill when x = 4?

Something to think about

In worked example 2, why is x = −1 excluded, when the divisor (x + 1) ÷ (x2 + 3x) is a perfectly good number, 0, at x = −1?

One way to think about it

On its own the divisor is 0 at x = −1, which is a fine value. But dividing by it would mean dividing by zero, and after flipping, x + 1 is a bottom. So the zeros of the divisor's top are excluded as well as the zeros of every bottom.

Exit ticket

8. Which is (x − 1) ÷ (x + 6) ÷ ((x − 1) ÷ (x + 2)) in its simplest form?